Learn | Grasp Maths

Lesson overview

Number — powers and roots

An index (or power) tells us how many times to multiply a number by itself. For example, $3^4 = 3 \times 3 \times 3 \times 3 = 81$ . Zero power: any non-zero number to the power of 0 equals 1, so $5^0 = 1$ . Negative powers: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ . The index laws are: $a^m \times a^n = a^{m+n}$ (multiply: add powers), $\frac{a^m}{a^n} = a^{m-n}$ (divide: subtract powers), and $(a^m)^n = a^{mn}$ (power of a power: multiply powers).

Zero and negative powers

Evaluate $5^0$ , $2^{-2}$ and $3^{-1}$ .

Any number to the power of 0 is 1. Negative powers mean 'flip to a fraction and make the power positive'.

Using the multiply rule: $a^m \times a^n = a^{m+n}$

Simplify $x^3 \times x^5$ .

RuleWhen multiplying powers with the same base, add the exponents

Write out what this means: $(x \times x \times x) \times (x \times x \times x \times x \times x) = x \times x \times \ldots$ (8 times).

Using the divide rule: $\frac{a^m}{a^n} = a^{m-n}$

Simplify $\frac{y^7}{y^2}$ .

RuleWhen dividing powers with the same base, subtract the exponents

$\frac{y^7}{y^2} = \frac{y \times y \times y \times y \times y \times y \times y}{y \times y}$ . Cancel two $y$ 's: $y^5$ remains.

Worked example — simplify using multiple index laws

Simplify $\frac{(2a^3)^2 \times a^{-1}}{a^4}$ .

Apply the power rule to the bracket: $(2a^3)^2 = 2^2 \times (a^3)^2 = 4a^6$ .
Multiply the numerator: $4a^6 \times a^{-1} = 4a^{6-1} = 4a^5$ (using the multiply rule).
Divide by the denominator: $\frac{4a^5}{a^4} = 4a^{5-4} = 4a$ (using the divide rule).
Check: does the final answer make sense? Yes — $4a$ is a simple linear expression.

Try it

Use index laws to simplify each expression. Show which rule you are using.

Question 1

What is $6^0$ ?

💡 Any non-zero number to the power of 0 equals 1.

Question 2

What is $2^{-3}$ as a fraction?

💡 $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ .

Question 3

Simplify $a^4 \times a^{-2}$ .

💡 When multiplying, add the exponents: $4 + (-2) = 2$ .

Question 4

Simplify $\frac{x^6}{x^3}$ .

💡 When dividing, subtract the exponents: $6 - 3 = 3$ .

Question 5

Evaluate $(3^{-2})^2$ . Give your answer as a fraction in simplest form or a decimal.

💡 Use the power rule: $(3^{-2})^2 = 3^{-4}$ . Then $3^{-4} = \frac{1}{81} = 0.0123...$

Common mistakes

Watch for these when working through the lesson.

Confusing zero power with zero value: $5^0 = 1$ , not $0$ .
Applying negative to the base: $-2^3 = -8$ (the negative is outside the power), but $(-2)^3 = -8$ (only when brackets are shown).
Adding exponents when dividing instead of subtracting: $\frac{a^5}{a^2} = a^3$ (subtract), not $a^7$ (add).

Indices including positive, negative and zero powers

Lesson overview

Worked example — simplify using multiple index laws

Try it

Common mistakes

Related topics

Independent practice will plug in here